Boost's Linear Algebra Solution for y=Ax

Question

Does boost have one? Where A, y and x is a matrix (sparse and can be very large) and vectors respectively. Either y or x can be unknown.

I can't seem to find it here: http://www.boost.org/doc/libs/1_39_0/libs/numeric/ublas/doc/index.htm

Answer 1

yes, you can solve linear equations with boost's ublas library. Here is one short way using LU-factorize and back-substituting to get the inverse:

using namespace boost::ublas;

Ainv = identity_matrix<float>(A.size1());
permutation_matrix<size_t> pm(A.size1());
lu_factorize(A,pm)
lu_substitute(A, pm, Ainv);

So to solve a linear system Ax=y, you would solve the equation trans(A)Ax=trans(A)y by taking the inverse of (trans(A)A)^-1 to get x: x = (trans(A)A)^-1Ay.

Answer 2

Linear solvers are generally part of the LAPACK library which is a higher level extension of the BLAS library. If you are on Linux, the Intel MKL has some good solvers, optimized both for dense and sparse matrices. If you are on windows, MKL has a one month trial for free... and to be honest I haven't tried any of the other ones out there. I know the Atlas package has a free LAPACK implementation but not sure how hard it is to get running on windows.

Anyways, search around for a LAPACK library which works on your system.

Answer 3

One of the best solvers for Ax = b, when A is sparse, is Tim Davis's UMFPACK

UMFPACK computes a sparse LU decomposition of A. It is the algorithm that gets used behind the scenes in Matlab when you type x=A\b (and A is sparse and square). UMFPACK is free software (GPL)

Also note if y=Ax, and x is known, but y is not, you compute y by performing a sparse matrix vector multiply, not by solving a linear system.